MinT - Best Known (165−87, 165, s)-Nets in Base 8

Best Known (165−87, 165, s)-Nets in Base 8

(165−87, 165, 130)-Net over F₈ — Constructive and digital

Digital (78, 165, 130)-net over F₈, using

t-expansion [i] based on digital (76, 165, 130)-net over F₈, using
- 7 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(165−87, 165, 214)-Net over F₈ — Digital

Digital (78, 165, 214)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−87, 165, 6682)-Net in Base 8 — Upper bound on s

There is no (78, 165, 6683)-net in base 8, because

1 times m-reduction [i] would yield (78, 164, 6683)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12806 842728 876130 404697 781985 396647 669323 850182 048442 506442 461726 114687 289842 902194 844549 095666 507499 303922 780961 755991 131103 148150 741121 146506 078592 > 8¹⁶⁴ [i]